Notation for knots

Notations are developed to describe diagrams of knots in a succinct
but useful way. The ultimate goal is to encode the knot diagram in a
way that enables someone else to reconstruct an equivalent diagram
from the code.

Gauss Code

The Gauss code for knot diagrams is perhaps the
easiest to describe. Suppose we have a diagram for some oriented
knot. Arbitrarily, pick a point P on the knot that is not on a
crossing. Then follow the orientation of the knot to arrive at the
first crossing, and label it 1. Then, follow the strand to the next
crossing. If the crossing you arrive at is not already labeled, then
label it 2 (i.e. one greater the crossing you last labeled). Else,
skip this crossing and proceed to the next one. Repeat this procedure
until all the crossings are labeled (once).

Given this labeling of the crossings, the notation is derived by
walking the knot again, starting at P. As we follow the knot, we
record the crossings we encounter, keeping in mind that if we happen
to traverse a crossing by an under-crossing strand, then
we record the label of the crossing with a negative sign. The
procedure terminates when you walk the whole knot. For 12 crossing knots a second sign convention is used: if the crossing is right handed it is given a positive sign, otherwise a negative sign.

For the example diagram, the Gauss code will be:

1 -2 3 -4 5 6 -7 -8 4 -9 2 -10 8 11 -6 -1 10 -3 9
-5 -11 7

Extended Gauss Code

In general, the Gauss Code for some knot diagram cannot reconstruct an
equivalent diagram, but a minor revision of the notation will make the
reconstruction possible. The revision is called the Extended Gauss
Code.

The crossings are labeled exactly as before. And, the crossings are
recorded as before except that rules for assigning the signs are
revised slightly. The first time we encounter a given crossing, we do
as before--over-crossing is positive, under-crossing is negative. The
second time we encounter a crossing, we assign a sign depending the
handedness of the crossing. If it is right-handed, then we assign
positive; if it is left-handed, negative.

Therefore, for the example, the Extended Gauss Code will be:

1 -2 3 -4 5 6 -7 -8 -4 -9
-2 -10 8 11 6 -1 -10 3 9 -5 -11 -7

The red numbers denote the numbers encoding the handedness of the
crossings.

DT Notation

The Dowker-Thistlethwaite Notation (DT Notation) of a given knot diagram can be
found with a similar procedure of labeling the crossings.

Pick an arbitrary point P on the knot that is not on a crossing. Then
follow the orientation of the knot to arrive at the first crossing,
and label it 1. Proceed to the next crossing and label it 2. One
might come across crossings that have been labeled already, but if
this is the case, give it a second label. (You will never need to
assign a third label, and eventually, every crossing will have two
labels.)

One observes that each crossings has an even label and an odd
label. The DT notation is derived by locating the odd labels in order (i.e. 1, 3, 5, etc...)
and recording the corresponding even labels, the sign of which is
determined by whether it was an over-crossing or an under-crossing
strand that assigned the odd label. If the odd label was assigned by
an under-crossing strand, we record the corresponding even label with
a negative sign.